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To find \( f'''(x) \), the third derivative of the function \( f(x) = -\frac{5}{6}x^3 + x^2 + \frac{3}{7} - \frac{1}{3}x^5 \), we need to differentiate \( f(x) \) three times. Let’s proceed with each derivative step-by-step.

1. **First Derivative \( f'(x) \):**
   \[
   f(x) = -\frac{5}{6}x^3 + x^2 + \frac{3}{7} - \frac{1}{3}x^5
   \]
   Differentiate each term:
   - The derivative of \( -\frac{5}{6}x^3 \) is \( -\frac{5}{6} \cdot 3x^2 = -\frac{5}{2}x^2 \).
   - The derivative of \( x^2 \) is \( 2x \).
   - The derivative of \( \frac{3}{7} \) (a constant) is \( 0 \).
   - The derivative of \( -\frac{1}{3}x^5 \) is \( -\frac{1}{3} \cdot 5x^4 = -\frac{5}{3}x^4 \).

   So,
   \[
   f'(x) = -\frac{5}{2}x^2 + 2x - \frac{5}{3}x^4
   \]

2. **Second Derivative \( f''(x) \):**
   Differentiate \( f'(x) \):
   - The derivative of \( -\frac{5}{2}x^2 \) is \( -\frac{5}{2} \cdot 2x = -5x \).
   - The derivative of \( 2x \) is \( 2 \).
   - The derivative of \( -\frac{5}{3}x^4 \) is \( -\frac{5}{3} \cdot 4x^3 = -\frac{20}{3}x^3 \).

   So,
   \[
   f''(x) = -5x + 2 - \frac{20}{3}x^3
   \]

3. **Third Derivative \( f'''(x) \):**
   Differentiate \( f''(x) \):
   - The derivative of \( -5x \) is \( -5 \).
   - The derivative of \( 2 \) (a constant) is \( 0 \).
   - The derivative of \( -\frac{20}{3}x^3 \) is \( -\frac{20}{3} \cdot 3x^2 = -20x^2 \).

   So,
   \[
   f'''(x) = -5 - 20x^2
   \]

**Final Answer:**
\[
f'''(x) = -20x^2 - 5
\]

Would you like further details on each step, or do you have any questions?

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Here are some related questions to deepen your understanding:

1. What is the fourth derivative of \( f(x) \) based on this answer?
2. How would \( f'''(x) \) change if \( f(x) \) included higher-degree terms, like \( x^6 \)?
3. Can you explain why constants disappear when taking derivatives?
4. How does the power rule apply to each term in \( f(x) \) when differentiating?
5. What would happen to \( f'(x) \) if you only differentiated the constant term \( \frac{3}{7} \)?

**Tip:** Always remember that each differentiation reduces the degree of each term by one.

Given the function f(x) = -5/6x^3 + x^2 + 3/7 - 1/3x^5, find f'''(x) in simplified form.

Explore how to find the third derivative of the function f(x) = -5/6x^3 + x^2 + 3/7 - 1/3x^5. This problem involves concepts of calculus, including the power rule for derivatives and higher-order derivatives of polynomial functions, making it ideal for students in Grades 11-12.

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